A variant of Newton's method is used to solve the resulting
system of nonlinear equations. Newton's method solves a system of
nonlinear equations, 
, by
successively applying updates in the direction of
This involves calculation of a residual, which can be calculated
directly from any of 4.24,
4.28, 4.29, and
4.30, and an approximation to the tangent
( 
). The best approximation to the tangent is the
Frechet derivative,
For the nonlinear systems given by 4.24,
4.29, and 4.30, the tangent matrix is
simply the sum of tangent of the static residual,
, and a scaled mass matrix.
Convergence of Newton methods can be determined using a variety of approaches. One method measures the magnitude of residual. The nonlinear method has converged if
or
Another convergence criteria measures the magnitude of the relative change in the full update. The nonlinear method has converged if
